Plotting Observed and Fitted

Let us do Figure 2.5, a plot of change versus setting identifying the level of program effort corresponding to each point. I will also superimpose the three parallel lines corresponding to the fitted model.

We call fitted() to calculate fitted values, use the levels of effort_g to label the points, left-justified with a bit of space, and then plot the fitted regression lines for each level of effort.

> library(ggplot2)
> fpe <- mutate(fpe, fitted=fitted(m2cg))
> png(file="fig25r.png", width=500, height=400) 
> ggplot(fpe, aes(setting, change, label=effort_g)) + geom_point() + 
+   geom_text(hjust="left", nudge_x=0.6, size=3) +
+   geom_line(data=group_by(fpe, effort_g), aes(setting, fitted, color=effort_g)) +
+   coord_cartesian(xlim=c(35, 95))
> dev.off()

png 
  2 

Adjusted and Unadjusted Means

Let us turn to the comparison of adjusted and unadjusted declines in Table 2.26, a useful way to present regression results to a non-technical audience.

> b <- coef(m2cg)   
> fpe <- mutate(fpe, adj_change = 
+   b[1] + b[2]*72.1 + b[3]*(effort_g=="Moderate") + b[4]*(effort_g=="Strong") )

Next we tabulate our data by level of effort and summarize observed and adjusted change.

> group_by(fpe, effort_g) |> 
+   summarize(observed=mean(change), adjusted=mean(adj_change))

# A tibble: 3 × 3
  effort_g observed adjusted
  <fct>       <dbl>    <dbl>
1 Weak         5        6.25
2 Moderate     9.33    10.4 
3 Strong      27.9     25.7 

Countries with strong program average a 28% decline in fertility, but they also tend to have higher settings; we estimate a slightly smaller decline of about 26% at average social setting. The estimate is based on the model, which adjusts linearly for setting and assumes that the slope is the same at all levels of effort. The next step will be to examine this assumption.

The Assumption of Parallelism

We will now allow the linear relationship between change and setting to vary with level of effort, by introducing an interaction between setting and the indicators of effort. Before we do that we center the index of social setting by subtracting the mean, a practice I highly recommend to simplify the interpretation of “main” effects when the model has interactions:

> fpe <- mutate(fpe, setting_c = setting - mean(setting))

We can now run the regression using

> m3cg <- lm(change ~ setting_c * effort_g, data = fpe)
> summary(m3cg)

Call:
lm(formula = change ~ setting_c * effort_g, data = fpe)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.7364 -2.2036 -0.0549  1.8122 11.4571 

Coefficients:
                           Estimate Std. Error t value Pr(>|t|)  
(Intercept)                 6.35583    2.47730   2.566   0.0224 *
setting_c                   0.18357    0.13970   1.314   0.2099  
effort_gModerate            3.58373    3.66235   0.979   0.3444  
effort_gStrong             13.33320    8.20916   1.624   0.1266  
setting_c:effort_gModerate -0.08684    0.23258  -0.373   0.7145  
setting_c:effort_gStrong    0.45670    0.60392   0.756   0.4620  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.959 on 14 degrees of freedom
Multiple R-squared:  0.8124,    Adjusted R-squared:  0.7454 
F-statistic: 12.13 on 5 and 14 DF,  p-value: 0.0001112

> anova(m3cg)

Analysis of Variance Table

Response: change
                   Df  Sum Sq Mean Sq F value    Pr(>F)    
setting_c           1 1201.08 1201.08 33.8263 4.476e-05 ***
effort_g            2  923.43  461.71 13.0034 0.0006426 ***
setting_c:effort_g  2   28.59   14.30  0.4026 0.6760530    
Residuals          14  497.10   35.51                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Compare the parameter estimates with Table 2.27 in the notes. You also have all the information required to produce the hierarchical anova in Table 2.28.

Because we centered setting, the coefficients for moderate and strong programs summarize differences by effort at mean setting, rather than at setting zero (which is well outside the range of the data). Thus, fertility decline averages 13 percentage points more under strong than under weak programs in countries with average social setting.

The interaction terms can be used to compute how these differences vary as we move away from the mean. For example in countries which are ten points above the mean social setting, the strong versus weak difference is almost five percentage points more than at the mean. These differences, however, are not significant, as we can’t reject the hypothesis that the three slopes are equal.

Exercise. Plot the data and the regression lines implied by the model with an interaction between linear setting and level of effort. Note how the difference between strong and weak programs increases with social setting. The interaction is not significant, however, so we have no evidence that the lines are not in fact parallel.

Updated fall 2022